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Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture

"I like to tell the stories of mathematics." With those words to a packed Carriage House audience on Sept. 19, 2007, Macalester College Professor and MAA President-Elect David M. Bressoud began to tell one of his favorite mathematical tales: the story of the alternating sign matrix conjecture. Audio

Bressoud's story revolved around the search for proofs of a variety of conjectures associated with n x n alternating sign matrices. Each such square matrix contains only the elements 0, 1, and -1, with the conditions that the sum of the entries in each row and each column is 1 and that the non-zero entries of each row and each column alternate in sign.

Robbins, Rumsey, and William Mills initially looked at the question of how many alternating square matrices there are. They found intriguing patterns and conjectured that certain formulas give the total number of n x n alternating sign matrices. But they were unable to prove their conjectures.

The trio "did what any really good mathematician does," Bressoud said. "You ask for help." So, Robbins, Rumsey, and Mills sought advice from Richard P. Stanley of the Massachusetts Institute of Technology. Audio

Stanley had been in contact with George Andrews of Pennsylvania State University earlier the same year because Andrews was working on a proof of his own and wanted Stanley's input. As it happens, Andrews had encountered exactly the same formula that Mills, Robbins, and Rumsey proposed for enumerating alternating sign matrices, but in an entirely different context: descending plane partitions.

That unexpected connection initiated much further work on alternating sign matrices, bringing in the ideas of other mathematicians and suggesting new conjectures.

The final breakthrough came from Greg Kuperberg of the University of California, Davis. He discovered that physicists working in statistical mechanics had been studying alternating sign matrices for years, in the context of a theoretical model they called square ice. Square ice is a two-dimensional arrangement of water molecules, with oxygen atoms at the vertices of a square lattice and one hydrogen atom between each pair of adjacent oxygen atoms. By applying methods that physicists used, it was possible to enumerate some of the quantities in which mathematicians were interested.

"Suddenly physicists became interested in the problems the mathematicians were working on, and mathematicians became interested in the problems the physicists were working on," Bressoud said.

Taking advantage of these and other insights, Doron Zeilberger of Rutgers University finally provided proofs of the key conjectures in the 1990s.

Bressoud's lecture vividly illustrated how research in mathematics actually progresses. It showed, too, how the work of individual mathematicians can quickly become intertwined with the work of countless others, often veering in surprising directions. "When you are looking for the result you think is out there, you will often encounter unexpected insights into seemingly unrelated problems," Bressoud noted.

And the story of alternating sign matrices continues. "There just is an incredible amount of work being done in this field today," Bressoud said. "This was the launching point for a lot of very exciting mathematics." Audio